Consequences of Observational Uncertainties on the Detection of Cosmic Topology

It is well known that as a consequence of local nature of general relativity, the global topology of space-time remains undetermined by the Einstein’s field equations. This coupled with the enormous recent increase in high resolution cosmological observations has led to a great deal of interest in the possibility that the universe may possess compact spatial sections with a non-trivial topology (see for example Refs. 2 and 3). These observations have also shown that the spatial curvature is very small (and possibly 0).8 Whatever the nature of cosmic topology may turn out to be, the issue of its detectability is of fundamental importance. Motivated by these observational results, a study was recently made of the question of detectability of the cosmic topology in nearly flat universes. It was demonstrated that as Ω0 → 1 increasing families of possible topologies become undetectable by methods based on image (or pattern) repetitions (see Refs. 4 – 6). However, measurements of the density parameters unavoidably involve observational uncertainties, and therefore any study of the detectability of the cosmic topology should take such uncertainties into account. In a recent paper,1 we studied the sensitivity of the detectability of cosmic topology to the uncertainties in the density parameters, using two complementary methods. Here we briefly summarise some of those results. As in standard cosmology we assume the universe is modelled by a 4-manifold M = R × M , with a locally isotropic and homogeneous Robertson-Walker (RW) metric, with a matter-energy content well approximated by dust (of density ρm) plus a cosmological constant Λ, with associated fractional densities Ωm = 8πGρm / (3H ) and ΩΛ ≡ Λ c / (3H), and Ω0 = Ωm + ΩΛ. We also assume a small but non-zero curvature, since a flat universe has no preferred length scale, and therefore the